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Spin-phonon coupling effects in transition-metal perovskites:

a DFT+U and hybrid-functional study

Jiawang Hong,1, ∗ Alessandro Stroppa,2 Jorge Íñiguez,3 Silvia Picozzi,2 and David Vanderbilt1

1 Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019, USA2CNR-SPIN, L’Aquila, Italy

3Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra, Spain(Dated: September 20, 2018)

Spin-phonon coupling effects, as reflected in phonon frequency shifts between ferromagnetic (FM)and G-type antiferromagnetic (AFM) configurations in cubic CaMnO3, SrMnO3, BaMnO3, LaCrO3,LaFeO3 and La2(CrFe)O6, are investigated using density-functional methods. The calculations arecarried out both with a hybrid-functional (HSE) approach and with a DFT+U approach using a Uthat has been fitted to HSE calculations. The phonon frequency shifts obtained in going from theFM to the AFM spin configuration agree well with those computed directly from the more accurateHSE approach, but are obtained with much less computational effort. We find that in the AMnO3materials class with A=Ca, Sr, and Ba, this frequency shift decreases as the A cation radius increasesfor the Γ phonons, while it increases for R-point phonons. In LaMO3 with M=Cr, Fe, and Cr/Fe,the phonon frequencies at Γ decrease as the spin order changes from AFM to FM for LaCrO3 andLaFeO3, but they increase for the double perovskite La2(CrFe)O6. We discuss these results and theprospects for bulk and superlattice forms of these materials to be useful as multiferroics.

PACS numbers: 75.85.+t, 75.47.Lx, 63.20.dk, 77.80.bg

I. INTRODUCTION

Multiferroic materials are compounds showing coexis-tence of two or more ferroic orders, e.g., ferroelectric-ity together with some form of magnetic order such asferromagnetism or antiferromagnetism. Presently suchmaterials are attracting enormous attention due to theirpotential for advanced device applications and becausethey offer a rich playground from a fundamental physicspoint of view. Possible applications tend to focus on themagnetoelectric (ME) coupling, which may pave the wayfor control of the magnetization by an applied electricfield in spintronic devices,1,2 although other applicationssuch as four-state memories3 are also of interest.Although intrinsic multiferroic materials are highly de-

sirable, they are generally scarce. One often-proposedreason may be that partially filled 3d shells favor mag-netism, while the best-known ferroelectric (FE) per-ovskites have a 3d0 configuration for the transition metal(e.g. Ti4+, Nb5+, etc.4). However, it has recentlybeen shown5 that some magnetic perovskite oxides dis-play incipient or actual FE instabilities, clearly indi-cating that there are ways around the usually-invokedincompatibility between ferroelectricity and magnetism.Some of these compounds, namely CaMnO3, SrMnO3and BaMnO3, will be considered in this work.Even if several microscopic mechanisms have been

identified for the occurrence of ferroelectricity in mag-netic materials,6,7 alternative routes are needed in or-der to optimize materials for functional device applica-tion. This could be done either by focusing on newclasses of compounds such as hybrid organic-inorganicmaterials8–10 or by modifying already-known materialsto engineer specific properties. In the latter direction,an intriguing possibility is to start with non-polar ma-

terials and then induce multiple non-polar instabilities;under appropriate circumstances this can produce a fer-roelectric polarization, as first predicted in Ref. 11 basedon general group theory arguments and analyzed in theSrBi2Nb2O9 compound by means of a symmetry analysiscombined with density-functional theory calculations byPerez-Mato et al.12 Here the ferroelectricity was found toarise from the interplay of several degrees of freedom, notall of them associated with unstable or nearly-unstablemodes. In particular, a coupling between polarizationand two octahedral-rotation modes was invoked to ex-plain the behavior.12 Bousquet et al. have demonstratedthat ferroelectricity is produced by local rotational modesin a SrTiO3/PbTiO3 superlattice.

13 Although in mostimproper ferroelectrics a single primary order parame-ter induces the polarization,14 Benedek and Fennie pro-posed that the combination of two lattice rotations, nei-ther of which produces ferroelectric properties individ-ually, can induce a ME coupling, weak ferromagnetism,and ferroelectricity.15 Indeed, we now know that rota-tions of the oxygen octahedra, in combination15–17 andeven individually,18,19 can produce ferroelectricity, mod-ify the magnetic order, and favor magnetoelectricity.

Another route to creating new multiferroic materialsmay be to exploit the coupling between polarization,strain, and spin degrees of freedom. A strong dependenceof the lowest-frequency polar phonon frequency on epi-taxial strain20 is common in paraelectric (PE) perovskiteoxides, and can sometimes be exploited to drive thesystem ferroelectric, a phenomenon known as epitaxial-strain–induced ferroelectricity.21 In a magnetic systemthat also has a strong spin-phonon coupling, i.e., a strongdependence of the polar phonon frequencies on spin or-der, the magnetic order may be capable of tipping thebalance between PE and FE states. For example, con-

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sider a system that has two competing ground states, oneof which is antiferromagnetic (AFM) and PE while theother is ferromagnetic (FM) and FE, and assume that thespin-phonon coupling is such that the lowest-frequencypolar phonon is softer for FM ordering than for AFMordering. In such a case, the epitaxial-strain enhance-ment of a polar instability may lead to a lowering of theenergy of the FM-FE state below that of the AFM-PEphase. Spin-phonon mechanisms of this kind have beenpowerfully exploited for the design of novel multiferroicsin Refs. 22–28.

Clearly, it is desirable to have a large spin-phonon cou-pling, in terms of a large shift ∆ω of phonon frequen-cies upon change of the magnetic order. Interestingly,recently computed spin-phonon couplings in ME com-pounds seems to be strikingly large. For example, a∆ω of about 60 cm−1 has been reported for the dou-ble perovskite La2NiMnO6.

29 Furthermore, ∆ω values ofabout 200 cm−1 have been computed for a cubic phaseof SrMnO3.

30 These values appear to be anomalouslylarge when compared to phonon splittings across mag-netic phase transitions in other oxides, which are in the5-30 cm−1 range.31,32

In the search for materials with large spin-phonon cou-pling, first-principles based calculations play a prominentrole since they can pinpoint promising candidates simplyby inspecting the dependence of the phonon frequencieson the magnetic order. However, a serious bottleneckappears. To obtain a reliable and accurate descriptionof these effects, it is important to describe the struc-tural, electronic and magnetic properties on an equalfooting. This is especially true for transition-metal ox-ides, which are the usual target materials for the spin-phonon driven ferroelectric-ferromagnet. In this case, itis well known that the localized nature of the 3d elec-tronic states, or loosely speaking, the “correlated” natureof these compounds, poses serious limits to the appli-cability of common density-functional methods like thelocal density approximation (LDA) or generalized gradi-ent approximation (GGA). In fact, these standard ap-proximations introduce a spurious Coulomb interactionof the electron with its own charge, i.e., the electrostaticself-interaction is not entirely compensated. This causesfairly large errors for localized states (e.g., Mn d states).It tends to destabilize the orbitals and decreases theirbinding energy, leading to an overdelocalization of thecharge density.33

One common way out is the use of the DFT+Umethod,34 where a Hubbard-like U term is introducedinto the DFT energy functional in order to take corre-lations partially into account. The method usually im-proves the electronic-structure description, but it suffersfrom shortcomings associated with the U -dependence ofthe calculated properties.35 Unfortunately, there is usu-ally no obvious choice of the U value to be adopted;common choices are usually based either on experimen-tal input or are derived from constrained DFT calcula-tions, but neither of these approaches is entirely satis-

factory. When dealing with phonon calculations, the U -dependence becomes even more critical since the phononfrequencies depend strongly on the unit-cell volume usedfor their evaluation, and the theoretical volume, in turn,depends on U . It is worth mentioning that even if anappropriate choice of U can accurately reproduce thebinding energy of localized d states of transition-metaloxides, it is by no means guaranteed that the same Ucan accurately reproduce other properties of the samecompound, such as the volume.35,36 While most papersaddress the spin-phonon coupling by applying DFT+Umethods, only a few deal with the dependence upon theU parameter.37 In this paper, we will show that the spin-phonon coupling can strongly depend on the U parame-ter, and that such a dependence may give rise to artifi-cially large couplings. It is important to stress this mes-sage, often overlooked in the literature, in view of theincreasing interest in ab-initio predictions of ferroelectricmaterials driven by spin-phonon coupling.

In the last few years, another paradigmatic approachhas been widely applied in solid-state materials science,namely the use of “hybrid functionals” that incorporate aweighted mixture of the exchange defined in the Hartree-Fock theory (but using the Kohn-Sham orbitals) and thedensity-functional exchange. The correlation term is re-tained from the density-functional framework. It is nowwidely accepted that the hybrid functionals outperformsemilocal functionals, especially for bulk materials withband gaps.38–49 It has also been shown that for low-dimensional systems such as semiconductor/oxides in-terfaces, the performance of hybrid functionals remainsquite satisfactory.50 However, some doubts have very re-cently been put forward about the performance of hybridfunctionals for certain structural configurations, e.g., sur-faces or nanostructures.51

While there is a plethora of different hybrid function-als, many of them defined empirically, a suitable func-tional derived on theoretical grounds is the so calledPBE0,52,53 where the exchange mixing parameter as beenfixed to one quarter as justified by a perturbation-theorycalculation. A closely related functional, the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional,54 intro-duces yet another parameter µ which splits the Coulombinteraction kernel into short- and long-range pieces whileretaining the mixing only on the short-range component.It has been shown that this new hybrid functional,54

while preserving most of the improved performance ofPBE0 with respect to standard local and semi-local ex-change correlation functionals, greatly reduces the com-putational cost. For this reason, it is especially suitablefor periodically extended systems, and is currently be-ing applied in many solid-state applications ranging fromsimple semiconductor systems to transition metals, lan-thanides, actinides, molecules at surfaces, diluted mag-netic semiconductors, and carbon nanostructures (for arecent review see Ref. 55). The HSE functional has beenalso used for phonon calculations for simple semiconduct-ing systems56,57 or perovskite structures,58,59 where it

3

was shown the that phonon modes are much more accu-rately reproduced using the hybrid functionals than usingGGA or LDA.60

Very recently, extended benchmarkings of the HSEmethod as well as other self-interaction–corrected ap-proaches have been presented for prototypical transi-tion metal oxides such as MnO, NiO and LaMnO3,and it has been shown that “HSE shows a remarkablequantitative agreement with experiments on most ex-amined properties”61 and “HSE shows predictive powerin describing exchange interactions in transition metaloxides.”62 These recent studies further motivate us touse the HSE functional for our studies of the spin-phononcoupling; indeed, as we will see in Sec. III B, the HSE-calculated phonon modes agree well with available exper-imental results for SrMnO3. The spin-phonon couplingeffect, to the best of our knowledge, is totally unexploredby hybrid functional approaches. In this work we aim atfilling this gap.

Even when using the more efficient HSE functional,however, the use of hybrids entails an increased compu-tational cost which makes the calculation of phonon prop-erties of complex magnetic oxides very difficult. In thispaper, we propose to circumvent this bottleneck by com-bining the HSE and DFT+U approaches, i.e., choosingthe appropriate U for each material by fitting to the HSEresults for some appropriate materials properties. Thisprovides a fairly efficient and affordable strategy thatpreserves the “HSE accuracy” for lattice constants, spin-phonon couplings, and related properties, while takingadvantage of the computationally inexpensive DFT+Umethod for the detailed calculations. Further details willbe given in Section II.

As far as the materials are concerned, it has been sug-gested that AMnO3 perovskites with A=Ca, Sr andBa may be good candidates for spin-phonon–couplingdriven multiferroicity.5,30,63 Furthermore, it has been re-ported that both SrMnO3 and CaMnO3 have a largespin-phonon coupling.30,64 We will show below that thespin-phonon coupling can depend strongly on the chosenU . We will also consider the simple perovskite LaMO3materials with M =Cr and Fe, which have Néel temper-atures above room temperature65 and large band gaps.We want to explore the possibility of using these twomaterials as building blocks for room temperature mul-tiferroics, e.g., in the form of double perovskites such asLa2CrFeO6. We have chosen these two classes of ma-terials in part for the reasons outlined above, but alsobecause they are sufficiently “easy-to-calculate” for thebenchmark and testing purposes of the present work, es-pecially when considering hybrid functionals.

The paper is organized as follows. In Sec. II we re-port the computational details and describe our strategyfor fitting U of the GGA+U calculations via a prelim-inary “exploratory” study using the HSE method. InSec. III we discuss: i) the effect of U on the frequencyshift, in Sec. III A; ii) the spin-phonon coupling effects inAMnO3 with A=Ca, Sr, and Ba, in Sec. III B; iii) the

spin-phonon coupling effects in LaMO3 with M =Cr, Fe,and the corresponding double perovskite La(Cr,Fe)O3 inSec. III C; and iv) prospects for the design of new multi-ferroics in Sec. III D. In Sec. IV, we give a summary andconclusions. Finally, in the Appendix, we give detailsabout the methodology used here for a full HSE phononcalculation.

II. METHODS

FIG. 1: (Color online) ABO3 perovskite structure doubledalong the [111] direction. A atoms (largest) shown in green;B atoms shown in violet; O atoms (smallest) shown in red.

All of our calculations were performed in the frame-work of density-functional theory as implemented inthe Vienna ab-initio simulation package (VASP-5.2)66,67

with a plane-wave cutoff of 500 eV. The DFT+U andHSE calculations were carried out using the same set ofprojector-augmented-wave (PAW) potentials to describethe electron-ion interaction.68,69 The unit cell for sim-ulating the G-AFM magnetic ordering, where all first-neighbor spins are antiferromagnetically aligned, is dou-bled along the [111] direction as shown in Fig. 1. Thesame simulation cell is retained for the FM configura-tion in order to avoid numerical artefacts that could arisein comparing calculations with different effective k-pointsamplings. A 6× 6× 6 Γ-centered k-point mesh is used.The phonon frequencies were calculated using the

frozen-phonon method. Except for the case ofLa2CrFeO6, the charge density and dynamical matrix re-tain the primitive 5-atom-cell periodicity for both the G-AFM and FM spin structures, so it is appropriate to an-alyze the phonons using symmetry labels from the prim-itive Pm3̄m perovskite cell. The zone-center phononsdecompose as 3 Γ−4 ⊕Γ

−

5 (plus acoustic modes), with theΓ−4 modes being polar. The zone-boundary modes at theR point (which appear at the Γ point in our 10-atom-cellcalculations) decompose as R+5 ⊕R

−

5 ⊕ 2R−

4 ⊕R−

2 ⊕R−

3 ,

4

FIG. 2: (Color online) Three idealized polar modes and R−5 mode in simple perovskites. (a) Slater mode; (b) Last mode; (c)Axe mode; (d) R−5 mode.

where the three-fold degenerate R−5 is of special inter-est because it corresponds to rigid rotations and tilts ofthe oxygen octahedra. After each frozen-phonon calcu-lation of the zone-center phonons of our 10-atom cell, weanalyze the modes to assign them according to these k-point and symmetry labels. For La2CrFeO6, some of themodes mix (e.g., Γ−4 with R

+5 ); in these cases we report

the dominant mode character.As for the polar Γ−4 phonons, these are often char-

acterized in ABO3 perovskites in terms of three kindsof idealized modes as illustrated in Figs. 2a-c. TheSlater mode (S-Γ−4 ) of Fig. 2(a) describes the vibrationof B cations against the oxygen octahedra; the Last (L-Γ−4 ) mode of Fig. 2(b) expresses a vibration of the Acations against the BO6 octahedra; and the Axe mode(A-Γ−4 ) of Fig. 2(c) represents the distortion of the oxy-gen octahedra.70 The actual mode eigenvectors never be-have precisely like these idealized cases, but we find thatthey can be identified in practice as being mainly ofone character, which is the one we report. These po-lar modes contribute to the low-frequency dielectric con-stant, and their softening in the high-symmetry paraelec-tric phase indicates the presence of a ferroelectric insta-bility. For the insulating compounds, we further calcu-lated their dielectric constants and Born effective chargesusing density-functional perturbation theory within theDFT+U context as implemented in VASP. The antifer-rodistortive (AFD) mode (R−5 mode), which describesthe rotation oxygen octahedra, is also shown in Fig. 2(d).We make use of the DFT+U method34 in the standard

Dudarev implementation where the on-site Coulomb in-teraction for the localized 3d orbitals is parametrized byUeff = U−J (which we denote henceforth as simply U)

71

using the PBEsol functional,72 which has been shown togive a satisfactory description of solid-state equilibriumproperties. We shall refer to this as PBEsol+U. The lackof experimentally available data for our systems preventsus from extracting U directly from experiments; we willreturn to this delicate point shortly.The other functional we have used is HSE06,54 a

screened hybrid functional introduced by Heyd, Scuse-ria, and Ernzerhof (HSE), where one quarter of thePBE short-range exchange is replaced by exact exchange,while the full PBE correlation energy is included. The

range-separation parameter µ is set to µ = 0.207 Å−1.The splitting of the Coulomb interaction into short- andlong-range pieces, as done in HSE, allows for a fasternumerical convergence with k-points when dealing withsolid-state systems. However, as previously mentioned,the application of the HSE approach to phonon calcula-tions for magnetic oxide systems remains very challengingin terms of computational workload.

Scheme to fit U from hybrid calculations

Here, we propose a practical scheme to perform rel-atively inexpensive DFT+U simulations that retain anaccuracy comparable to HSE for the calculation of spin-phonon couplings.Our goal is to obtain a DFT+U approach that re-

produces the dependence of the materials properties onthe spin arrangement as obtained from HSE calculations.Naturally, the first and most basic property we wouldlike to capture correctly is the energy itself. As it turnsout, the energy is also a very sensible property on whichone can base a U -fitting scheme: The energy differencesbetween spin configurations are directly related to themagnetic interactions or exchange constants, and theseare known to depend on the on-site Coulomb repulsionU affecting the electrons of the magnetic species. (Typi-cally, in our compounds of interest, the value of U used inthe simulations will play an important role in determin-ing the character of the top valence states; in turn, thiswill have a direct impact on the nature and magnitudeof the exchange couplings between spins.) This depen-dence of the exchange constants on U makes this crite-rion a very convenient one for our purposes. Of course,such a fitting procedure does not guarantee our DFT+Uscheme will reproduce correctly the phonon frequenciesand frequency shifts between different spin arrangementsobtained from HSE calculations. In that sense, we arerelying on the physical soundness of the Hubbard-U cor-rection to DFT; as we will see below, the results are quiteconvincing.Obtaining U from the energy differences has an addi-

tional advantage: It allows us to devise a very simple fit-ting procedure that relies on a minimal number of HSE

5

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

0 1 2 3 4 5 6

∆E

(eV

)

U (eV)

∆EHSE∆E+U

∆Ex+U

FIG. 3: (Color online) Variation of PBEsol+U total energydifference ∆E = EAFM − EFM with U for LaCrO3, ∆E

+U

is the energy difference with volume fixed to the optimizedvolume from HSE and ∆E+Ux relaxed volume for each U . (Thefitted value of U occurs at the crossing with the ∆EHSE value.)

calculations. In essence, these are the steps we follow.We relax the cubic structure separately for G-AFM andFM spin configurations using HSE and obtain the total-energy difference ∆E(HSE) =EAFM (HSE) - EFM(HSE).We then carry out a series of PBEsol+U calculations inwhich U is varied from 0 to 6 eV and obtain ∆E(+U) =EAFM(+U)−EFM(+U) for each U . The U is then chosesuch that ∆E(+U)= ∆E(HSE).

Figure 3 shows the results of our fitting for LaCrO3by using two slightly different approaches. In one casewe ran the PBEsol+U simulations at the HSE-optimizedvolumes, and in the other case we performed volume re-laxations at the PBEsol+U level for each U value consid-ered. It can be seen that such a choice does not have a bigeffect on the computed frequency shifts, and both resultin the same U value. Hence, the U values reported herewere obtained by running PBEsol+U simulations at theHSE volumes.73 We obtained U= 3.0, 2.8, 2.7, 3.8, and5.1 eV, respectively, for CaMnO3, SrMnO3, BaMnO3,LaCrO3 and LaFeO3.

74 We then do detailed phonon cal-culations using this value of U in the PBEsol+U cal-culations to investigate spin-phonon coupling effects inAMnO3 and LaMO3.

As we will show in Sec. III, we have tested this pro-posed scheme and found that it works quite well. In par-ticular, the phonon frequency shifts ∆ω = ωAFM − ωFMcomputed using PBEsol+U with the fitted U are almostthe same as those obtained using HSE directly. Since aprevious study has concluded that HSE works “remark-ably well” for transition-metal oxides,61 we believe thisapproach can be used with confidence.

Finally, we note that the direct HSE calculations canbe rather heavy, even though we only have 10-atom cells;the presence of magnetic order and the need to calculatethe phonons and the spin-phonon couplings makes thecalculations challenging.75 We circumvent this difficulty

-100

0

100

200

300

400

500

0 2 4 6 8

∆ω

(cm

-1)

U (eV)

(a) S-Γ4-

L-Γ4-

A-Γ4-

R5-

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0 2 4 6 8

ω2

(104

cm

-2)

U (eV)

(b)

AFM-S-Γ4-

AFM-R5-

FM-S-Γ4-

FM-R5-

FIG. 4: (Color online) Effect of U on spin-phonon couplingin SrMnO3. (a) Γ

−

4 and R−

5 phonon frequency shifts (∆ω =ωAFM−ωFM) versus U . (b) S-Γ

−

4 and R−

5 phonon frequenciesversus U for AFM and FM states.

by splitting a single run of HSE frozen-phonon calcula-tions into several parallel runs, each one computing theforces for symmetry-independent atomic displacements.We then use the forces calculated in these runs to con-struct the force-constant matrix, and by diagonalizingthis matrix, we obtain the phonon frequencies and eigen-vectors. Further details of this procedure are presentedin the Appendix.

III. RESULTS AND DISCUSSION

A. Effect of U on the frequency shifts

Let us begin by showing some representative resultsof the U -dependence of the phonon frequencies and fre-quency shifts for the AMnO3 and LaMO3 compounds.Here we focus on the case of SrMnO3, which has beenpredicted to exhibit a large spin-phonon coupling basedon GGA+U simulations using U= 1.7 eV; more precisely,a giant frequency shift ∆ω= 230 cm−1 has been reportedfor the Slater mode at the theoretical equilibrium state.30

We computed the phonon frequencies and frequencyshifts of the ideal cubic perovskite phase of SrMnO3 us-ing U values in the range between 0 and 8 eV. Figure 4

6

TABLE I: Lattice constant a, local Mn magnetic moment m,and band gap Egap (zero if blank) for CaMnO3, SrMnO3 andBaMnO3 from PBEsol+U (+U) and HSE methods.

a (Å) m (µB) Egap(eV )

CaMnO3-AFM+U 3.73 2.79 0.35HSE 3.73 2.81 1.70

CaMnO3-FM+U 3.74 2.84HSE 3.73 2.81

SrMnO3-AFM+U 3.80 2.80 0.40HSE 3.80 2.85 1.45

SrMnO3-FM+U 3.81 2.89HSE 3.81 2.91 0.00a

BaMnO3-AFM+U 3.91 2.86 0.13HSE 3.91 2.92 1.20

BaMnO3-FM+U 3.93 3.14HSE 3.93 3.13

aHalf-metallic.

shows our results for the key modes that determine theoccurrence of ferroelectricity, i.e., the Γ−4 polar phononsand the R−5 antiferrodistortive (AFD) mode. The R

−

5

mode involves antiphase rotations of the O6 octahedraaround the three principal axes of the perovskite lattice;such a mode is soft in many cubic perovskites, and it of-ten competes with the FE instabilities to determine thenature of the low-symmetry phases.From Fig. 4a we see that the Γ−4 and R

−

5 modes de-pend strongly on the chosen U . Notably, the frequencyshift ∆ω of the S-Γ−4 mode can change from 400 cm

−1

to 0 cm−1 as U increases from 0 to 8 eV. The ∆ω of theA-Γ−4 and R

−

5 modes also depend on the U , while theL-Γ−4 mode is nearly insensitive to U within this rangeof values. Interestingly, the large frequency shift of S-Γ−4is related to the strong U -dependence of the FM S-Γ−4mode, as shown in Fig. 4b.Hence, our results show that the magnitude of the

spin-phonon coupling has a significant dependence on thevalue of U employed in a DFT+U calculation, and thatsuch a dependence is particularly strong for some of thekey modes determining the structural (FE/AFD) insta-bilities of cubic perovskite oxides. It is thus clear thatchoosing an appropriate U is of critical importance if wewant to avoid artificially “strong” couplings.

B. Spin-phonon coupling in CaMnO3, SrMnO3,

and BaMnO3

We studied the cubic phase of CaMnO3, SrMnO3,and BaMnO3 using the PBEsol+U approximation withU values for Mn’s 3d electrons (i.e., U = 3.0 eV forCaMnO3, U = 2.8 eV for SrMnO3, and U = 2.7 eVfor BaMnO3) that were determined as described in Sec-tion II. The basic properties that we obtained are listedin Table I. Our PBEsol+U method gives good lattice

TABLE II: Phonon frequencies and frequency shifts calcu-lated for CaMnO3, in cm

−1. Frequencies calculated usingHSE and PBEsol+U : ωHSE and ω+U . Frequency differencebetween HSE and PBEsol+U : ω∆ = ω+U −ωHSE. Frequencyshift: ∆ω = ωAFM − ωFM. Difference of frequency shifts be-tween HSE and PBEsol+U : ∆ω∆ = ∆ω+U −∆ωHSE.

ωHSEAFM ω∆AFM ω

HSEFM ω

∆FM ∆ω

HSE ∆ω∆

L-Γ−4 126 –29 -81 –18 207 –10S-Γ−4 272 –11 186 –3 86 –8Γ−5 241 –48 183 –72 58 24R−3 658 10 636 11 22 –1R−4 430 –46 413 –49 17 2R−5 –204 –18 –218 –20 14 2A-Γ−4 569 35 557 40 12 –5R+5 437 –11 448 –14 –11 3R−2 890 –27 885 –31 5 4R−4 160 –20 157 –21 3 1

constants and local Mn magnetic moments comparedwith HSE. As regards the metallic or insulating char-acter, the PBEsol+U agrees qualitatively with the HSEresult in most cases, although the band gap is under-estimated. In FM-SrMnO3 there is a clear discrepancybetween PBEsol+U and HSE: the HSE calculations pre-dict a half-metallic state, while we obtain a metal withPBEsol+U.

We have also calculated the Born effective chargesfor insulating AFM configurations of CaMnO3 (SrMnO3,BaMnO3). We obtain ZCa/Sr/Ba=2.60 e (2.58 e, 2.73 e),ZMn=7.35 e (7.83 e, 9.42 e), ZO‖=−6.55 e (−6.93 e,

−8.07 e), ZO⊥=−1.70 e (−1.74 e, −2.04 e), where e is theelementary charge and ZO‖ and ZO⊥ refer, respectively,to the dynamical charges defined for an atomic displace-ment parallel and perpendicular to the Mn-O bond. Theanomalously large ZMn and ZO‖ charges of AFM stateof CaMnO3, SrMnO3 and BaMnO3 are strongly reminis-cent of the results obtained for ferroelectric perovskiteoxides,76 and suggest the possible proximity of a polarinstability that might be triggered by an appropriate ex-ternal (e.g., strain) field.5,30,63

The phonons at Γ and R for different spin orders forCaMnO3, SrMnO3 and BaMnO3 are shown in Tables II,III, and IV, respectively; the phonon modes are orderedby descending ∆ωHSE. The first thing to note from thetable is that the frequencies obtained for SrMnO3’s po-lar modes (i.e., 177, 187 and 494 cm−1) agree well withavailable low temperature experiment results (i.e., 162,188 and 498 cm−1 taken from Ref. 77). Thus, our resultsprovide additional evidence that the HSE scheme rendersaccurate phonon frequencies for magnetic oxides. Thesecond thing to note from these tables is that the phonon-frequency shifts (∆ω) obtained with PBEsol+U are inoverall good agreement with the HSE results, which sug-gests that our strategy to fit the value of U is a goodone.

7

TABLE III: Calculated phonon frequencies and frequencyshifts, as in Table II, but for SrMnO3.

ωHSEAFM ω∆AFM ω

HSEFM ω

∆FM ∆ω

HSE ∆ω∆

S-Γ−4 177 40 –96 –16 273 56R−5 64 –135 –6 –121 70 –14Γ−5 292 –38 253 –54 39 16R−2 811 –26 789 –25 22 –1R−3 571 12 553 2 18 10R−4 424 –47 410 –62 14 15L-Γ−4 187 –16 177 –7 10 –9A-Γ−4 494 22 487 17 7 5R+5 412 –12 409 –12 3 0R−4 159 –6 157 –7 2 0

Third, our results offer information about the depen-dence of the structural instabilities on the choice of theA-site cation and on the magnetic arrangement. We findthat the AFD instability (R−5 mode) becomes weaker anddisappears, for both AFM and FM states, as the size ofthe A-site cation increases (The effective ionic radii esti-mated by Shannon78 are rCa = 1.34 Å, rSr = 1.44 Å, andrBa = 1.61 Å.) In contrast, the ferroelectric instability(Slater mode) becomes stronger as the A-site cation be-comes bigger. This is the usual behavior that one wouldexpect for these two instabilities of the cubic perovskitestructure, and has been recently examined in detail byother authors.5,79,80 It clearly suggests that some of thesecompounds could display a magnetically ordered ferro-electric ground state. In particular, this could be thecase for SrMnO3 and BaMnO3: in these compounds, theAFD instability becomes weaker or even disappears, sothat it can no longer compete with and suppress the FEsoft mode.

The phonon frequency shift ∆ω between the AFMand FM magnetic orders is shown in Fig. 5. We

-50

0

50

100

150

200

250

300

S-Γ4- Γ5

- L-Γ4-A-Γ4

- R3- R4

- R4- R5

+ R2- R5

-

∆ω

(cm

-1)

CaMnO3SrMnO3BaMnO3

FIG. 5: (Color online) Frequency shifts ∆ω = ωAFM − ωFMin AMO3 from HSE calculations.

TABLE IV: Calculated phonon frequencies and frequencyshifts, as in Table II, but for BaMnO3.

ωHSEAFM ω∆AFM ω

HSEFM ω

∆FM ∆ω

HSE ∆ω∆

S-Γ−4 –274 117 –369 227 95 –110R−2 670 –28 631 –59 39 31Γ−5 317 –34 281 –28 36 –6R+5 363 –13 335 –22 28 9R−5 221 –28 200 –67 21 39R−4 403 –48 384 –60 19 12A-Γ−4 423 –12 404 11 19 –22R−3 423 16 407 –20 16 36L-Γ−4 200 –6 195 –6 5 0R−4 150 –5 147 –6 3 1

find that the Slater modes exhibit a very consider-able spin-phonon coupling for all three compounds, with∆ω & 100 cm−1. A sizable effect is also obtained forthe Γ−5 mode (∆ω ∼ 50 cm

−1) in all cases. Additionally,the Last mode shows a very large effect for CaMnO3,and the spin-phonon coupling for R−5 is significant in thecase of SrMnO3. The coupling is relatively small, or evennegligible, for all other modes.

We will not attempt any detailed interpretation of ourquantitative results here, but some words are in order.In the recent literature, it has often been claimed that

(a)

(b)

FIG. 6: (Color online) Changes to metal–O–metal bond angle(solid blue line) resulting from Slater (a) and Γ−5 (b) phononmodes. Open and solid dots indicate ideal and displaced po-sitions respectively. Small red dots are oxygen; larger purpledots are metal atoms.

8

TABLE V: Lattice constant a, local magnetic moment m, band gap Egap, dielectric constant ε, and Born effective charge Zfor LaCrO3, LaFeO3 and La2CrFeO6 (LCFO) from GGA+U (+U) and HSE methods.

a (Å) m (µB) Egap (eV) ε∞ ε0 ZLa (e) ZCr/Fe (e) ZO‖ (e) ZO⊥ (e)

LaCrO3-AFM+U 3.87 2.86 2.01 6.12 93.71 4.53 3.34 −3.50 −2.19HSE 3.87 2.75 3.06

LaCrO3-FM+U 3.88 2.84 1.70 6.08 236.19 4.52 3.61 −3.62 −2.26HSE 3.88 2.74 2.49

LaFeO3-AFM+U 3.90 4.19 2.20 6.37 4.47 3.75 −3.41 −2.43HSE 3.91 4.10 3.25

LaFeO3-FM+U 3.91 4.38 1.57 6.19 4.48 4.02 −3.59 −2.46HSE 3.93 4.26 2.20

LCFO–AFM+U 3.88 2.71/4.30 1.98 6.35 713.57 4.51 3.02/4.22 −3.50 −2.33HSE 3.88 2.63/4.19 2.78

LCFO–FM+U 3.89 3.01/4.28 2.01 6.18 262.40 4.49 3.21/3.87 −3.45 −2.30HSE 3.89 2.89/4.18 3.04

AFD distortions are expected to couple strongly withthe magnetic structure of perovskite oxides, as they con-trol the metal–oxygen–metal angles that are critical todetermine the magnitude of the magnetic interactionsthat are dominant in insulators.15 [The hopping param-eters between oxygen (2p) and transition metal (3d) or-bitals are strongly dependent on such angles, as are theeffective hoppings between 3d orbitals of neighboringtransition metals. As we know form the Goodenough-Kanamori-Anderson rules, the super-exchange interac-tions are strongly dependent on such hoppings.] In prin-ciple, such an effect should be reflected in our computedfrequency shifts; for such R−5 modes we obtain ∆ω valuesranging between 10 cm−1 and 70 cm−1. Interestingly, fol-lowing the same argument, one would conclude that theSlater and Γ−5 modes should have a similar impact onthe magnetic couplings in these materials, as both involvechanges in the metal–oxygen–metal angles due to the rel-ative displacement of metal and oxygen atoms, as shownin Fig. 6. Further, the Slater modes also affect the metal–oxygen distances, as they involve a significant shorteningof some metal–oxygen bonds (Fig. 6a). Hence, from thisperspective, the large ∆ω values obtained in our calcula-tions for these two types of modes are hardly surprising.Finally, the result obtained for the Last mode in the caseof CaMnO3 (i.e., ∆ω ≈ 200 cm

−1) is clearly anomalousand unexpected according to the above arguments. Wespeculate that it may be related to a significant Ca–Ointeraction that alters the usual super-exchange mecha-nism and favors a FM interaction. While unusual, effectsthat seem similar have been reported previously for othercompounds.81,82

C. Spin-phonon coupling in LaCrO3, LaFeO3, and

La2CrFeO6

In the previous Section we have shown how spin-phonon coupling effects can crucially depend on the na-ture of the A-site cations in AMnO3 compounds. Now

we will focus on the change of B-site cation to investigatethe spin-phonon couplings in the LaMO3 compounds(M=Cr, Fe) and the double perovskite La2CrFeO6. Forour PBEsol+U calculations we used U = 3.8 eV for Crand U = 5.1 eV for Fe, which were determined as de-scribed in Section II. We obtained these U ’s from calcu-lations for LaCrO3 and LaFeO3 and used the same valuesfor the PBEsol+U study of La2CrFeO6.

The basic computed properties of LaCrO3, LaFeO3and La2CrFeO6 are presented in Table V. In all cases, weobtain insulating solutions for AFM and FM spin orders,both from HSE and PBEsol+U calculations. Therefore,we also calculated the Born effective charges and opticaldielectric constants within PBEsol+U by using density-functional perturbation theory as implemented in VASP.The static dielectric constants were also calculated forcompounds which do not have an unstable polar mode.It is evident that ZLa and ZO⊥ are essentially identicalfor the three compounds and insensitive to the spin or-der. On the other hand, the ZCr/Fe and ZO‖ chargesof LaCrO3 and LaFeO3 increase in magnitude when thespin arrangement changes from AFM to FM, and de-

TABLE VI: Calculated phonon frequencies and frequencyshifts, as in Table II, but for LaCrO3.

ωHSEAFM ω∆AFM ω

HSEFM ω

∆FM ∆ω

HSE ∆ω∆

Γ−5 230 –24 178 –35 52 11S-Γ−4 361 –26 317 –27 44 1L-Γ−4 69 0 48 –3 21 3R−4 389 –29 372 –30 17 1R+5 414 2 430 0 –16 2A-Γ−4 659 32 644 30 15 2R−5 –213 1 –228 1 15 0R−3 683 5 669 5 14 0R−4 89 –4 81 –4 8 0R−2 840 –17 843 –19 –3 2

9

crease in the case of the double perovskite La2CrFeO6.Table V also shows that the optical dielectric constantsε∞ are very close to 6 for these three materials, and areindependent of the magnetic order. However, the staticdielectric constants ε0 of LaCrO3 and La2CrFeO6 changevery significantly when moving from AFM to FM. Also,the static dielectric constant of AFM-La2CrFeO6 is verylarge due to the very small frequency of the Last phononmode. The static dielectric constants of LaFeO3 are notshown in Table V because the Last modes are unstable,and strictly speaking they are not well defined. (Roughlyspeaking, in all such cases we would have ε0 → ∞, as thecubic phase is unstable with respect to a polar distor-tion.)

In Tables VI and VII we show the phonon frequen-cies as calculated at the HSE and PBEsol+U levels forLaCrO3 and LaFeO3 respectively. Further, Table VIIIshows the PBEsol+U results for La2CrFeO6. As was thecase for the AMnO3 compounds of the previous section,we find here as well that the AFM-FM frequency shiftscomputed with our PBEsol+U scheme reproduce well theHSE results.

Our results also show that the phonons of the LaMO3compounds exhibit some features that differ from thoseof the AMnO3 materials. First, the octahedral rotationmode (R−5 ) is unstable for all the La-based compounds,and it is largely insensitive to the nature of the B-sitecation. Second, the energetics of the FE modes is verydifferent. In the case of the Mn-based compounds, theSlater mode is the lowest-frequency mode, and in somecases it becomes unstable, thus inducing a polarization,just as the unstable Slater mode induces ferroelectricityin BaTiO3.

70 However, in the La compounds the lowest-frequency mode is the Last phonon mode, and in thecases of LaFeO3 and La2CrFeO6 this Last mode is un-stable and might induce ferroelectricity; this situation ismore analogous to what occurs in PbTiO3

70 or BiFeO3.

The spin-phonon coupling effects computed for theLaMO3 compounds are given in Tables VI, VII, and VIII,and are summarized in Fig. 7. Here, the first thing tonote is that the magnitude of the effects is significantly

TABLE VII: Calculated phonon frequencies and frequencyshifts, as in Table II, but for LaFeO3.

ωHSEAFM ω∆AFM ω

HSEFM ω

∆FM ∆ω

HSE ∆ω∆

R−2 796 –38 822 –40 -26 2A-Γ−4 645 –7 630 –4 15 –3S-Γ−4 259 4 247 4 12 0R−3 549 –11 536 –8 13 –3R−4 390 –24 379 –23 11 –1L-Γ−4 –81 12 –92 9 11 3R−5 –237 9 –247 7 10 2R−4 66 –2 56 –2 10 0R+5 346 –5 355 –6 –9 1Γ−5 120 –3 115 –10 5 7

-30

-20

-10

0

10

20

30

40

50

60

Γ5- S-Γ4

-L-Γ4-A-Γ4

- R4- R5

- R4- R3

- R2- R5

+

∆ω

(cm

-1)

LaCrO3LaFeO3

LCFO

FIG. 7: (Color online) Frequency shifts for LaMO3 (HSE forLaCrO3 and LaFeO3; PBEsol+U for La2CrFeO6).

smaller than for the LaMO3 compounds (note the dif-ferent scales of Figs. 5 and 7). Second, we observe thatthe largest effects are associated with the Γ−5 and Slatermodes (with ∆ω ≈ 45 cm−1 for LaCrO3). This is consis-tent with the point made above that these modes disruptthe metal–oxygen–metal super-exchange paths. In com-parison, in this case we obtain a relatively small effect forthe R−5 modes, which show ∆ω values (. 20 cm

−1) thatare comparable to those computed for most of the phononmodes considered. Finally, for LaCrO3 and LaFeO3 weobserve that the Γ phonon frequencies decrease as thespin order changes from AFM to FM, in line with whatwas observed for the AMnO3 compounds. In contrast,the Γ modes of double-perovskite La2CrFeO6 increase infrequency when the spin order changes to FM.

D. Prospects for the design of new multiferroics

Let us now discuss several possible implications of ourresults regarding the design of novel multiferroic materi-

TABLE VIII: Calculated phonon frequencies and frequencyshifts, as in Table II, but for La2CrFeO6 from PBEsol+U .

83

ωAFM ωFM ∆ωL-Γ−4 31 45 –14Γ−5 161 178 –17R+5 383 400 –17S-Γ−4 284 297 –13R−3 629 616 13R−2 803 797 6R−5 –225 –222 –3R−4 362 360 2A-Γ−4 674 676 –2R−4 79 79 0

10

als.

1. BaMnO3-based materials

As already mentioned in the Introduction, the spin-phonon coupling can trigger multiferroicity in some ma-terials if, by application of an epitaxial strain or otherperturbation, a FM-FE state can be stabilized with re-spect to the AFM-PE ground state.22 Indeed, the threeinvestigated AMnO3 compounds have been suggested ascandidates to exhibit strain-induced multiferroicity bythree research groups.5,30,63 According to our calcula-tions, we can tentatively suggest that BaMnO3 mightshow multiferroicity even without strain applied, pro-vided the material can be grown as a distorted perovskite(the most stable polymorph of BaMnO3 adopts instead astructure with face-sharing octahedra63,84). Note that itmay be possible to enhance the stability of BaMnO3’s cu-bic perovskite phase by partial substitution of Ba by Caor Sr. In fact, ideally one would try to obtain samples ofBa1−xSrxMnO3 or Ba1−xCaxMnO3 with x large enoughto stabilize the perovskite phase, and small enough forthe FE instability associated to the FM Slater mode todominate over the R−5 and AFM Slater modes. Appar-ently this strategy to obtain new multiferroics has re-cently been realized experimentally by Sakai et al.85 inBa1−xSrxMnO3 solid solutions. An alternative would beto consider CaMnO3/BaMnO3 or SrMnO3/BaMnO3 su-perlattices with varying ratios of the pure compounds,and perhaps tuning the misfit strain via the choice ofsubstrate.Another intriguing possibility pertains to the mag-

netic response of the AMnO3 compounds that displayan AFM-PE ground state and a dominant polar insta-bility of their FM phase. Again, this could be the casefor some Ba1−xSrxMnO3 and Ba1−xCaxMnO3 solid so-lutions with an appropriate choice of x. By applying amagnetic field to such compounds, it might be possibleto switch them from the AFM ground state to a FMspin configuration and, as a result, induce a ferroelectricpolarization.

2. Double-perovskite La2CrFeO6

The double perovskite La2CrFeO6 has been intensivelystudied to examine its possible magnetic order through3d3 − 3d5 superexchange. However, its magnetic groundstate has long been debated. Pickett et al.86 predictedthat the ferrimagnetic (FiM) ground state with a netspin moment of 2µB/f.u. is more stable than the FMone with 7µB/f.u. (This FiM order can be viewedas a G-AFM configuration in which, for example, allFe spins are pointing up and all Cr spins are point-ing down.) However, from GGA and LDA+U calcu-lations, Miura et al. found that the ground-state mag-netic ordering of La2CrFeO6 is FiM in GGA, but that

even a small U in LDA+U makes it FM.87 The exper-imental picture is also unclear. Ueda et al. have growna (111)-oriented LaCrO3/LaFeO3 superlattice which ex-hibits FM ordering, although the measured saturationmagnetization is much smaller than expected.88 Very re-cently, Chakraverty et al. reported epitaxial La2CrFeO6double-perovskite films grown by pulsed-laser deposition,and their sample exhibits FiM with a saturation magne-tization of 2.0± 0.15µB/f.u. at 5K.

89

Our HSE calculations for La2CrFeO6 with the atomic(oxygen) positions relaxed in the cubic structure showsthe magnetic ground state has a FiM spin pattern lead-ing to a FiM structure with a net magnetization of1.56µB/f.u. This is consistent with LDA

86 and GGA87

calculations, as well as being heuristically consistent withthe experimental report of AFM ordering in La2CrFeO6solid-solution films.89 However, our HSE calculationshows that the energy difference between the FiM andFM states is very small: FiM is only 0.8meV/f.u. lowerin energy than FM. In addition, the GGA+U with U fit-ted to LaCrO3 and LaFeO3 (U=3.8 and 5.1 eV for Cr andFe, respectively) results in a FM magnetic ground statehaving a total energy 34.9meV lower than that of theFiM. By doing a fitting of the U parameters for Cr andFe directly to EAFM−EFM of La2CrFeO6 as computed byPBEsol+U and HSE, instead of for LaCrO3 and LaFeO3separately, we find parameters of U=3.0 and 4.1 eV forCr and Fe respectively (the phonon properties predictedfrom these are very close to our previous results). Usingthese parameters, we find that the FiM ground state is0.7meV lower in energy than the FM state. For compar-ison, a straight PBEsol calculation (with U = 0) predictsthat the energy of the FiM ground state is 596meV/f.u.lower than that of the FM state. Clearly, U should bechosen carefully in order to obtain the correct groundstate of La2CrFeO6.

According to our HSE calculations, the magneticground state of La2CrFeO6 is FiM, but with the FMstate lying only very slightly higher in energy. This maybe the reason why questions about the magnetic groundstate of La2CrFeO6 have long been debated; the energydifference is so small that external perturbations (e.g.,the epitaxial strain in a superlattice88) or variations inU between different LDA+U87 and GGA+U calculationsmay bring the FM energy below that of the AFM. Takentogether with our results, shown in Table VIII, that theLast mode is close to going soft in this material, this sug-gests that La2CrFeO6 might be a good candidate for amaterial in which multiferroic phase transitions could beinduced, similar to what was shown for SrMnO3

30 andSrCoO3.

27 Because the spin ordering is so delicate, itseems likely that a small misfit strain could be enough totrigger such a transition.

11

IV. SUMMARY AND CONCLUSIONS

In this work, we have studied the spin-phonon couplingfor transition-metal oxides within density-functional the-ory. From the computational point of view, an accuratedescription of the electronic, structural, and vibrationalproperties on an equal footing is a prerequisite for a re-liable study of the coupling between spins and phonon.Taking note of the increasing evidence that hybrid func-tionals are suitable for this task, we have adopted theHeyd-Scuseria-Ernzerhof (HSE) screened hybrid func-tional for the present work. However, the accuracy ofthe HSE results comes at the cost of an increase of com-putational load, so that a full frozen-phonon calculationof the phonon modes remains prohibitively expensive inmany cases. We propose to overcome this limitation bycarrying out calculations at the DFT+U level using Uparameters that have been fitted to HSE results for total-energy differences between spin configurations. Our re-sults show that the resulting DFT+U scheme reproducesthe HSE results very accurately, especially in regard tothe spin-phonon couplings of interest here.

As regards the direct HSE phonon calculations, wehave developed an approach in which we split the calcu-lation into separate, simultaneous frozen-phonon calcula-tions for different symmetry-adapted displacement pat-terns, and then combine the results to calculate and di-agonalize the dynamical matrix, thus accelerating thesecalculations significantly.

Our important results can be summarized as follows.First, we have shown that the choice of U is a big concernin such studies, since the spin-phonon coupling can de-pend very strongly on U . As an alternative to extractingU from experimental studies, we propose here to obtain itby fitting to HSE calculations as illustrated above. Sec-ond, we have studied CaMnO3, SrMnO3 and BaMnO3,focusing on trends in the spin-phonon coupling due tothe increase of the A cation size. Based on the straincouplings and the spin-phonon interactions, we suggesttheoretically that BaMnO3 is more likely to show fer-roelectricity under tensile strain and, furthermore, thatA-site substitution by a cation with smaller size may in-duce multiferroicity even without external strain. Third,we find that in the AMnO3 materials class with A=Ca,Sr, and Ba, the frequency shift decreases as the A cationradius increases for the Γ phonons, while it increases forR-point phonons. Fourth, we have shown that chang-ing B-site cations may also have important effects on thedielectric properties: in LaMO3 with M=Cr, Fe, andCr/Fe, the phonon frequencies at Γ decrease as the spinorder changes from AFM to FM for LaCrO3 and LaFeO3,but they increase for the double perovskite La2(CrFe)O6.Finally, we have shown that the polar phonon modes ofthe investigated perovskites tend to display the largestspin-phonon couplings, while modes involving rotationsof the O6 octahedra present considerable, but generallysmaller, effects. Such observations may be relevant asregards current efforts to obtain large magnetostructural

(and magnetoelectric) effects.We hope that our study will stimulate further work

leading to rational design and strain engineering of mul-tiferroicity using spin-phonon couplings.

Appendix: Efficient phonon calculations with hybrid

functionals

Even though we only have a 10-atom cell, we foundthat it can be quite expensive to use the HSE functionalto carry out the needed spin-polarized calculations ofphonon properties.75 We overcome this limitation as fol-lows. First, we use symmetry to limit ourselves to setsof displacements that will block-diagonalize the force-constant matrix. For example, for the polar modes wemove the cations along x and the O atoms along x andy. We then carry out self-consistent calculations on thesedisplaced geometries, and from the forces we constructthe relevant block of the force-constant matrix. Second,while the standard VASP implementation uses a “centraldifference” method in which ions are displaced by smallamounts in both positive and negative displacements, wesave some further effort by displacing only in the posi-tive direction. Finally, we note that the forces resultingfrom each pattern of atomic displacements can be calcu-lated independently, allowing us to split the calculationin parallel across independent groups of processors andthus further reduce the wall-clock time.We have checked the accuracy of this approach for FM-

LaCrO3 using PBEsol+U (U=3.8 eV) and for G-AFM-BaMnO3 using HSE. For each case, we compared the re-sults of the standard implementation of the VASP frozen-ion calculation of phonon frequencies with the revised ap-proach described above. We take the ion displacementsto be 0.015 Å in all cases. We find that the RMS er-ror of ten different phonon frequencies is 1.2 cm−1 forPBEsol+U and 7.3 cm−1 for HSE. These results suggestthat this method has acceptable accuracy with reducedcomputational cost. We propose that it could be usedalso for cases of lower symmetry and larger cells, thusmaking the HSE phonon calculations at Γ affordable ingeneral.Recently, a revised Perdew–Burke–Ernzerhof func-

tional base on PBEsol, which we refer to as HSEsol, wasdesigned to yield more accurate equilibrium propertiesfor solids and their surfaces. Compared to HSE, signifi-cant improvements were found for lattice constants andatomization energies of solids.90 We also checked the ef-fect of using HSEsol on our calculations, as shown inTable IX. From this table, we can see that the phononscalculated with HSEsol are very close to those from HSE.

Acknowledgments

This work was supported by ONR Grant 00014-05-0054, by Grant Agreement No. 203523-BISMUTH of the

12

TABLE IX: Comparison of phonon frequencies (cm−1) ofSrMnO3 calculated from HSE and HSEsol (sol).

ωHSEAFM ωsolAFM ω

HSEFM ω

solFM ∆ω

HSE ∆ωsol

S-Γ−4 177 166 –96 –129 273 295Γ−5 292 290 253 250 38 40L-Γ−4 187 179 177 172 10 7A-Γ−4 494 485 487 477 7 8R−4 159 156 157 154 2 3R+5 412 408 409 405 3 3R−4 424 419 410 406 14 14R−3 571 558 553 539 18 19R−5 64 71 –6 26 70 45R−2 811 803 789 782 21 20

EU-FP7 European Research Council, and by MICINN-Spain (Grants No. MAT2010-18113, No. MAT2010-10093-E, and No. CSD2007-00041). Computations werecarried out at the Center for Piezoelectrics by Design.J.H. acknowledges travel support from AQUIFER Pro-grams funded by the International Center for MaterialsResearch at UC Santa Barbara. We thank Jun Hee Lee,Claude Ederer and Karin Rabe for useful discussions.

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mailto:[email protected]

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73 In the materials considered here, the equilibrium volumesobtained from HSE and PBEsol+U are essentially the same(see Tables I and V); hence, the determined U’s are essen-tially independent from our imposing the HSE-volume inthe PBEsol+U calculations or not. However, in the generalcase this may not be true, and it seems reasonable to per-form structural relaxations at the DFT+U level for eachU considered.

74 We found two U values, U = 2.2 and 5.1 eV, that generatethe same ∆E+U = ∆EHSE for LaFeO3. We chose 5.1 eVbecause the band gap and magnetic moment are in betteragreement with HSE calculations. We also found the FMenergy to cross below the AFM one for U < 0.8 eV.

75 In our case it takes about 6 days to obtain the phononfrequency for G-AFM BaMnO3 using a direct HSE calcu-lation with 96 cores (AMD Opteron 6136 2.4GHz with In-finiband network), but less then 0.5 hours using PBEsol+Uwith only 16 cores.

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−

4 , so we denote it as R+5 .

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